Minimal Retentive Sets in Tournaments From Anywhere to TEQ Felix - PowerPoint PPT Presentation

Minimal Retentive Sets in Tournaments – From Anywhere to TEQ – Felix Brandt Markus Brill Felix Fischer Paul Harrenstein Ludwig-Maximilians-Universität München Estoril, April 12, 2010 1 / 17

The Trouble with Tournaments Tournaments are oriented complete graphs Many applications: social choice theory, sports tournaments, game theory, argumentation theory, webpage and journal ranking, etc. Question: How to select the winner(s) of a tournament in the absence of transitivity? a c b e d 2 / 17

Overview Tournament solutions Retentiveness and Schwartz’s Tournament Equilibrium Set (TEQ) Properties of minimal retentive sets ‘Approximating’ TEQ A new tournament solution 3 / 17

Tournament Solutions a c b A tournament T = ( A , ≻ ) consists of: • a finite set A of alternatives • a complete and asymmetric relation ≻ on A e d 4 / 17

Tournament Solutions a c b A tournament T = ( A , ≻ ) consists of: • a finite set A of alternatives • a complete and asymmetric relation ≻ on A e d A tournament solution S maps each tournament T = ( A , ≻ ) to a set S ( T ) such that ∅ � S ( T ) ⊆ A and S ( T ) contains the Condorcet winner if it exists • S is called proper if a Condordet winner is always selected as only alternative 4 / 17

Tournament Solutions a c b A tournament T = ( A , ≻ ) consists of: • a finite set A of alternatives • a complete and asymmetric relation ≻ on A e d A tournament solution S maps each tournament T = ( A , ≻ ) to a set S ( T ) such that ∅ � S ( T ) ⊆ A and S ( T ) contains the Condorcet winner if it exists • S is called proper if a Condordet winner is always selected as only alternative Examples: Trivial Solution (TRIV), Top Cycle (TC), Uncovered Set, Slater Set, Copeland Set, Banks Set, Minimal Covering Set (MC), Tournament Equilibrium Set (TEQ) , . . . 4 / 17

Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) a b 5 / 17

Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) 5 / 17

Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) b a 5 / 17

Basic Properties of Tournament Solutions Monotonicity (MON) Weak Superset Property (WSP) Strong Superset Property (SSP) Independence of Unchosen Alternatives (IUA) b Note: a SSP is equivalent to ˆ α (see Felix’s lecture) (SSP ∧ MON) implies WSP and IUA 5 / 17

Examples Definition: TRIV returns the set A for each tournament T = ( A , ≻ ) 6 / 17

Examples Definition: TRIV returns the set A for each tournament T = ( A , ≻ ) Definition: TC returns the smallest dominating set , i.e. the smallest set B ⊆ A with B ≻ A \ B • Intuition: No winner should be dominated by a loser 6 / 17

Examples Definition: TRIV returns the set A for each tournament T = ( A , ≻ ) Definition: TC returns the smallest dominating set , i.e. the smallest set B ⊆ A with B ≻ A \ B • Intuition: No winner should be dominated by a loser B A \ B 6 / 17

Examples Definition: TRIV returns the set A for each tournament T = ( A , ≻ ) Definition: TC returns the smallest dominating set , i.e. the smallest set B ⊆ A with B ≻ A \ B • Intuition: No winner should be dominated by a loser • Define D ( b ) = { a ∈ A : a ≻ b } • TC is the smallest set B satisfying D ( b ) ⊆ B for all b ∈ B B D ( b ) b A \ B B 6 / 17

Examples Definition: TRIV returns the set A for each tournament T = ( A , ≻ ) Definition: TC returns the smallest dominating set , i.e. the smallest set B ⊆ A with B ≻ A \ B • Intuition: No winner should be dominated by a loser • Define D ( b ) = { a ∈ A : a ≻ b } • TC is the smallest set B satisfying D ( b ) ⊆ B for all b ∈ B Both TRIV and TC satisfy all four basic properties B D ( b ) b A \ B B 6 / 17

Retentiveness Intuition: • An alternative a is only “properly” dominated by a “good” alternatives Thomas Schwartz 7 / 17

Retentiveness Intuition: • An alternative a is only “properly” dominated by a “good” alternatives, i.e., alternatives selected by S from the dominators of a Thomas Schwartz 7 / 17

Retentiveness Intuition: • An alternative a is only “properly” dominated by a “good” alternatives, i.e., alternatives selected by S from the dominators of a • No winner should be “properly” dominated by a loser Thomas Schwartz 7 / 17

Retentiveness B is S-retentive if B � ∅ and S ( D ( b )) ⊆ B for Definition: all b ∈ B D ( b ) Thomas Schwartz b B 7 / 17

Retentiveness B is S-retentive if B � ∅ and S ( D ( b )) ⊆ B for Definition: all b ∈ B D ( b ) D ( b ) Thomas Schwartz b B 7 / 17

Retentiveness B is S-retentive if B � ∅ and S ( D ( b )) ⊆ B for Definition: all b ∈ B D ( b ) D ( b ) S ( D ( b )) Thomas Schwartz b B 7 / 17

Retentiveness B is S-retentive if B � ∅ and S ( D ( b )) ⊆ B for Definition: all b ∈ B D ( b ) D ( b ) S ( D ( b )) Thomas Schwartz b B ˚ Definition: S returns the union of all minimal S -retentive sets 7 / 17

Retentiveness B is S-retentive if B � ∅ and S ( D ( b )) ⊆ B for Definition: all b ∈ B D ( b ) D ( b ) S ( D ( b )) Thomas Schwartz b B ˚ Definition: S returns the union of all minimal S -retentive sets • Call ˚ S unique if there always exists a unique minimal S -retentive set 7 / 17

Retentiveness B is S-retentive if B � ∅ and S ( D ( b )) ⊆ B for Definition: all b ∈ B D ( b ) D ( b ) S ( D ( b )) Thomas Schwartz b B ˚ Definition: S returns the union of all minimal S -retentive sets • Call ˚ S unique if there always exists a unique minimal S -retentive set • Minimal S -retentive sets exist for each tournament • ˚ S is unique if and only if there do not exist two disjoint S -retentive sets 7 / 17

Example ˚ Proposition: TRIV = TC 8 / 17

Example ˚ Proposition: TRIV = TC Proof: A set is TRIV -retentive if and only if it is dominating D ( b ) TRIV ( D ( b )) = D ( b ) b B 8 / 17

The Tournament Equilibrium Set ˚ The tournament equilibrium set (TEQ) is defined recursively as TEQ = TEQ 9 / 17

Minimal Retentive Sets in Tournaments From Anywhere to TEQ Felix - PowerPoint PPT Presentation

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